A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

• URL: http://arxiv.org/abs/2106.02119v1
• Date: Thu, 3 Jun 2021 20:31:00 GMT
• Title: A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples
• Authors: Christian K\"ummerle, Claudio Mayrink Verdun
• Abstract summary: We propose an iterative algorithm for low-rank matrix completion. It is able to complete very ill-conditioned matrices with a condition number of up to $10$ from few samples.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.

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